For loop groups of the form \(C^\infty ({\mathbb {S}}^1,K)\) ,
\(K\)  a compact connected Lie group, the irreducible
projective positive energy representations can be identified
as highest weight representations ([1]},
[2]}, and in particular
[3]} for \(K = ).With the convexity theorems in \cite {AP83} and \cite {KP84} it shouldbe possible to show that these representationsare semibounded and one can hope for an analog ofTheorem~\ref {thm:8.22} asserting that every irreduciblesemibounded representation of the corresponding double extensioneither is a highest weight representation or its dual.In \cite {Ne09b} it is shown on the algebraic level that therealso exist many interesting unitary representationsof infinite rank generalizations of twisted and untwistedloop algebras, resp., their double extensions(so-called locally affine Lie algebras). On the group levelthey correspond to (double extensions of) groupsof the form \) C(S1, K)\(, where\) K\( is a Hilbert--Lie group, such as \) U2(H)\(.It seems quite likely that all these representationsare semibounded with momentum sets generated by a singlecoadjoint orbit.\)New sources of semibounded representations:
Although many interesting classes of semibounded representations
are known, a systematic understanding of the geometric sources
of these representations is still lacking.
As we have seen above, on the Lie algebra level the existence
of open invariant cones in a trivial central extension
is necessary (Remark REF ). However, as the example
\(\mathcal {V}({\mathbb {S}}^1)\)  shows, it is not sufficient
(Theorem REF ).
